The Furstenberg Poisson Boundary and CAT(0) Cube Complexes

Talia Fernós
Department of Mathematics and Statistics, University of North Carolina at Greensboro, 317 College Avenue, Greensboro, NC 27412, USA
t_fernos@uncg.edu

Abstract.

We show under weak hypotheses that ∂X, the Roller boundary of a finite dimensional CAT(0) cube complex X is the Furstenberg-Poisson boundary of a sufficiently nice random walk on an acting group Γ. In particular, we show that if Γ admits a nonelementary proper action on X, and μ is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a μ-stationary measure on ∂X making it the Furstenberg-Poisson boundary for the μ-random walk on Γ. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties.

The author was partially supported by NSF Grant number DMS-1312928, and UNCG New Faculty Summer Excellence Research Grant

CAT(0) cube complexes are fascinating objects of study, thanks in part to the interplay between two metrics that they naturally admit, the CAT(0) metric, and the median metric. Restricted to each cube, these coincide either with the standard Euclidean metric (ℓ2) or with the “taxi-cab” metric (ℓ1). Somewhat recently, CAT(0) cube complexes played a crucial roll in Agol’s proof of the Virtual Haaken Conjecture (an outstanding problem in the theory of 3-manifolds) [Ago13], [KM12], [HW08], [Wis09], [BW12]. Examples of CAT(0) cube complexes and groups acting nicely on them include trees, (universal covers of) Salvetti complexes associated to right angeled Artin groups, Coxeter Groups, Small Cancellation groups, and are closed under taking finite products.

Associated to a random walk on a group one has the Furstenberg-Poisson boundary. It is in some sense, the limits of the trajectories of the random walk. Its existence, as an abstract measure space, for a generating random walk is guaranteed by the seminal result of Furstenberg [Fur73]. This important object has since established itself as an integral part in the study of rigidity (see for example [BF14]) in particular by realizing it as a geometric boundary of the group in question.

One may associate to any CAT(0) space a visual boundary where each point is an equivalence class of geodesic rays. The visual boundary for a CAT(0) space gives a compactification of the space, at least when the space is locally compact [BH99]. For a wide class of hyperbolic groups, and more generally, certain groups acting on CAT(0) spaces, the visual boundary is a Furstenberg-Poisson boundary for suitably chosen random walks [Kai94], [KM99].

The wall metric naturally leads to the Roller compactification of a CAT(0) cube complex. Nevo and Sageev show that the Roller boundary (see Section 2.3) can be made to be a Furstenberg-Poisson boundary for a group Γ when the group admits a nonelementary proper co-compact action on X[NS13]. The purpose of this paper is to give a generalization of this result to groups which admit a nonelementary proper action on a finite dimensional CAT(0) cube complex. The complex is not assumed to be locally compact, and in particular, the action is not required to be co-compact. Our approach will be somewhat different to that of Nevo and Sageev and in particular, we will not address several of the dynamical questions that they consider: for example that the resulting stationary measure is unique, or that the action is minimal or strongly proximal. Such questions will be examined in a forthcoming paper by Lécureux, Mathéus, and the present author.

Let μ be a probability measure on a discrete countable group Γ. Assume that it is generating, i.e. that the semi-group generated by the support of μ is the whole of Γ. Recall that a probability measure μ on Γ is said to have finite entropy if

H(μ):=−∑γ∈Γμ(γ)logμ(γ)<∞.

Also, if |⋅|:Γ→R is a pseudonorm on Γ then μ is said to have finite first logarithmic moment (with respect to |⋅|) if ∑γ∈Γμ(γ)log|γ|<∞. (See Section 8.1 for more details.) If we have an action of Γ on X, then fixing a basepoint o∈X allows us to consider the pseudonorm defined by |γ|o:=d(γo,o).

Main Theorem.

Let X be a finite dimensional CAT(0) cube complex, Γ a discrete countable group, Γ→Aut(X) a nonelementary proper action by automorphisms on X, and μ a generating probability measure on Γ of finite entropy. If there is a base point o∈X for which μ has finite first logarithmic moment then there exists a probability measure ϑ on the Roller boundary ∂X such that (∂X,ϑ) is the Furstenberg-Poisson boundary for the μ-random walk on Γ. Furthermore, ϑ gives full measure to the regular points in ∂X.

The proof of the Main Theorem follows a standard path. We first show that the Roller Boundary is a quotient of the Furstenberg-Poisson boundary (Section 7) and then apply Kaimanovich’s celebrated Strip Condition to prove maximality (Section 8).

We note that Karlsson and Margulis show that the visual boundary of a CAT(0) space is the Furstenberg-Poisson boundary for suitable random walks [KM99]. They assume very little about the space, but assume that the measure μ has finite first moment and that orbits grow at most exponentially. The Main Theorem above applies to the restricted class spaces (i.e. CAT(0) cube complexes), which pays off by allowing for significantly weaker hypotheses on the action and the measure μ.

Observe that our Main Theorem applies for example to any non-elementary subgroup of a right angeled artin group or more generally of a graph product of finitely generated abelian groups [RW13].

Furthermore, we remark on the importance that the regular points are of full measure: they exhibit strong contracting properties. This will be exploited to study random walks on CAT(0) cube complexes in the forthcoming paper of Lécureux, Mathéus, and the present author mentioned above.

An action on a CAT(0) cube complex is said to be Roller nonelementary if every orbit in the Roller compactification is infinite (see Section 2.3). This notion guarantees nonamenability of the closure of the acting group in Aut(X), and characterizes it for X locally compact. This Tits’ alternative, is essentially an encapsulation of results of Caprace and Sageev [CS11], Caprace [CFI12], and Chatterji, Iozzi, and the author [CFI12]. It also comes after several versions of Tits’ alternatives (see [CS11], [SW05]). The statement is in the spirit of Pays and Valette [PV91]:

Theorem 1.1 (Tits’ Alternative).

Let X be a finite dimensional CAT(0) cube complex and Γ⩽Aut(X). If X is locally compact then the following are equivalent:

Γ does not preserve any interval I⊂¯¯¯¯¯X.

The Γ-action is Roller nonelementary.

Γ contains a nonabelian free subgroup acting freely on X.

The closure ¯¯¯¯Γ in Aut(X) is nonamenable.

Remark 1.2.

In fact, the condition that X be locally compact is only necessary for the implication (4) ⟹ (1). All the other implications, namely (1) ⟺ (2) ⟺(3) ⟹ (4) hold in the non-locally compact case as well.

Acknowledgements: The author is grateful to the following people for their kindness and generosity: Uri Bader, Greg Bell, Ruth Charney, Indira Chatterji, Alex Furman, Alessandra Iozzi, Vadim Kaimanovich, Jean Lécureux, Seonhee Lim, Amos Nevo, Andrei Malyutin, Frédéric Mathéus, and Michah Sageev. Conversations and collaborations with these people made this article possible. Further thanks go to the University of Illinois at Chicago, the Centre International de Rencontres Mathématique, and the Institut Henri Poincaré.

We will say that a metric space is a Euclidean cube if there is an n∈N for which it is isometric to [0,1]n with the standard induced Euclidean metric from Rn.

Definition 2.1.

A second countable finite dimensional simply-connected metric polyhedral complex X is a CAT(0) cube complex if the closed cells are Euclidean cubes, the gluing maps are isometries and
the link of each vertex is a flag complex.

Recall that a flag complex is a simplicial complex in which each complete subgraph on (k+1)-vertices is the 1-skeleton of a k-simplex in the complex. That the link of every vertex is a flag complex is equivalent to the condition of being locally CAT(0), thanks to Gromov’s Link Condition.

We remark that we absorb the condition of finite dimensionality in the definition of a CAT(0) cube complex and as such, we will not explicitly mention it in the sequel. Furthermore, if the dimension of the CAT(0) cube complex is D, then this is equivalent to the existence of a maximal dimensional cube of dimension D.

A morphism between two CAT(0) cube complexes is an
isometry that preserves the cubical structures, i.e. it is an isometry f:X→Y such that f(C) is a cube of Y whenever C is a cube in X. We denote by
Aut(X) the group of automorphisms of X to itself.

2.1. Walled Spaces

A space with walls or a walled space is a set S together with a countable collection of non-empty subsets H⊂2S called half-spaces with the following properties:

If h∈H then h≠∅.

There is a fixed-point free involution ∗:H→H

h↦h∗:=X∖h.

The collection of half-spaces separating two points of S is finite, i.e. for every p,q∈S the set of half-spaces h∈H such that p∈h and q∈h∗ is finite.

There is a D∈N such that for every collection of pairwise transverse half-spaces, {h1,…,hn} we must have that n⩽D.

A pair of half-spaces h,k∈H is said to be transverse if the following four intersections are all non-empty:

h∩k,h∩k∗,h∗∩k∗,h∗∩k.

Associated to a walled space is the wall pseudo-metric d:S×S→R:

d(p,q)=12#({h∈H:p∈h,q∈h∗}∪{h∈H:q∈h,p∈h∗}).

This satisfies the properties of a metric, with the exception that d(p,q)=0 does not necessarily imply that p=q.

Let us then consider the associated quotient S∼ consisting of equivalence classes of points of S whose pseudo-wall distance is 0. Clearly, the wall pseudo-metric descends to a metric on S∼.

For h∈H the wall associated to h is the unordered pair {h,h∗}. This explains the terminology, as well as the factor of 12 in the definition of the (pseudo-)wall metric.

2.2. CAT(0) Cube Complexes as Walled Spaces

As we shall now see, CAT(0) cube complexes naturally admit a walled (pseudo-)metric and are in some sense the unique examples of such spaces.

Let [0,1]n be an n-dimensional cube. The ith coordinate projection is denoted by pri:[0,1]n→[0,1]. A wall of a cube [0,1]n is the set pr−1i{1/2}. Observe that the complement of each wall in a cube has two connected components.

Definition 2.2.

A wall of a CAT(0) cube complex X is a convex subset whose intersection with each cube is either a wall of the cube or empty.

The complement of a wall in a CAT(0) cube complex has two connected components [Sag95, Theorem 4.10] which we call half-spaces and we denote them by H(X). Observe that since X is second countable, there are countably many half-spaces in H(X).

The notation and terminology here is purposefully chosen to remind the reader of a walled space. Indeed, in essence, a walled space uniquely generates a CAT(0) cube complex [Sag95], [CN05], [Nic04]. And it is this walled space structure of the CAT(0) cube complex that we will ultimately be interested in, if not fascinated by. Since walls separate points in the zero-skeleton of a CAT(0) cube complex, we will in fact consider the zero-skeleton as our object of study.

Let X0 denote the vertex set of X and H(X0)={h∩X0:h∈H(X)}.
This yields a fixed-point free involution ∗:H(X0)→H(X0)

(1)

h0↦h∗0:=X0∖h0.

One drawback of passing to the zero-skeleton, is that a wall is no longer a subset of X0. Therefore, for h0∈H(X0), we will denote by ^h0 the pair {h0,h∗0} and think of it as a wall, as in Section 2.1.

Let (S,H) be a walled metric space. Then, there exists a CAT(0) cube complex X and an embedding ι:S↪X0 such that:

If S and X0 are endowed with their respective wall metrics then ι is an isometry onto its image.

The set map induced by ι is a bijection H→H(X0), h↦k such that

k∩ι(S)=ι(h).

If γ:S→S is a wall-isometry then there exists a unique extension to an automorphism γ0:X0→X0 that agrees with γ on ι(S).

Furthermore, if (X0,H(X0)) is the walled space associated to the vertex set of a CAT(0) cube complex X, then the above association applied to (S,H)=(X0,H(X0)) yields once more X, and ι:X0→X0 can be taken to be the identity and the induced homomorphism Aut(X0)→Aut(X0) is the identity isomorphism.

When a collection of half-spaces H is given, we will denote the associated CAT(0) cube complex as X(H), leading to the somewhat abusive formulation of the last part of Theorem 2.3:

X(H(X0))=X.

This striking result shows that the combinatorial information of the wall structure completely captures the geometry of the CAT(0) cube complex. This will be exploited in what follows. To this end, we now set X=X0, and H=H(X0). Unless otherwise stated, every metric property will be taken with respect to the wall metric.

The first of many beautiful properties of CAT(0) cube complexes is a type of Helly’s Theorem:

Theorem 2.4.

Keeping with the terminology of transverse half-spaces introduced in Section 2.1, if h1,…,hn∈H are pairwise transverse half-spaces then n⩽D.

2.3. Roller Duality

Given a subset s⊂H of halfspaces,
we denote by s∗ the collection {h∗:h∈s}.
We say that s satisfies:

the totality condition if s∪s∗=H;

the consistency condition if, s∩s∗=∅ and if h∈s and h⊂k,
then k∈s.

Fix v∈X and consider the collection Uv={h∈H:v∈h}. It is straightforward to verify that Uv satisfies both totality and consistency as a collection of half-spaces.
Roller Duality is then obtained via the following observation:

∩h∈Uvh={v}.

This shows that if w∈X then we have that

Uv=Uw⟺v=w,

giving an embedding X↪2H obtained by v↦Uv. This embedding is made isometric by endowing 2H with the extended metric1

d(A,B)=12#(A△B).

For now, let us consider X⊂2H. Then, the Roller compactification is denoted by ¯¯¯¯¯X and is the closure of X in 2H. The Roller boundary is then ∂X=¯¯¯¯¯X∖X. Observe that in general, while ¯¯¯¯¯X is a compact space containing X as a dense subset, it is not a compactification in the usual sense. Indeed, unless X is locally compact, the embedding X↪¯¯¯¯¯X does not have an open image, and ∂X is not closed. This is best exemplified by taking the wedge sum of countably many lines. The limit of any sequence of distinct points in the boundary will be the wedge point. While it is also true that the visual boundary is not a compactification when X is not locally comapact, the Roller boundary does present one significant advantage: the union X⊔∂X is indeed compact.

With this notation in place, the partition {h,h∗} extends to a partition of ¯¯¯¯¯X and hence, when we speak of a half-space as a collection of points, we mean

h⊂¯¯¯¯¯X=h⊔h∗.

Remark 2.5.

Given h∈H, we denote the set {h,h∗} by ^h. By abuse of notation, for k∈H, we will say that ^h⊂k if and only if h⊊k or h∗⊊k. This is consistent with the standard notion of the wall corresponding to a mid-cube.

We now give characterizations of special types of subsets of ¯¯¯¯¯X. To this end, we say that s∈2H satisfies the descending chain condition if every infinite descending chain of half-spaces is eventually constant.

Facts 1.

The following are true for a non-empty s∈2H:

If s satisfies the consistency condition then

∅≠∩h∈sh⊂¯¯¯¯¯X.

If s satisfies the consistency condition and the descending chain condition then

∅≠(∩h∈sh)∩X.

The collection s satisfies both the totality and consistency conditions if and only if there exists v∈¯¯¯¯¯X such that s=Uv. Fixing Uv∈2H we have that

v∈X if and only if Uv satisfies the descending chain condition.

v∈∂X if and only if Uv contains a nontrivial infinite descending chain, i.e. for each n there is an hn∈S such that hn+1⊊hn.

Let us say a few words about why these facts are true, or where one can find proofs, though likely several proofs are available. In case of Item (1), this is simple if one can show that the collection has the finite intersection property as ¯¯¯¯¯X is compact. Furthermore, the CAT(0) cube complex version of Helly’s Theorem 2.4 allows one to pass from finite intersections to pairwise intersections, and this last case is easy to verify given the condition of consistency.
For the second item, we refer the reader to Lemma 2.3 of [NS13].
Finally, for the last item, we refer the reader to [Rol].

There are also other special sets which will be of interest:

Definition 2.6.

The collection of nonterminating elements is denoted by ∂NTX and consists of the elements v∈∂X such that every finite descending chain can be extended,
i.e. given h∈Uv there is a k∈Uv such that

k⊂h.

In general, it may be the case that ∂NTX is empty. However, in case X admits a nonelementary action (see Section 3.1) then ∂NTX is not empty [NS13], [CFI12].

2.4. The Median

The vertex set of a CAT(0) cube complex
with the edge metric (equivalently with the wall metric) is a median space [Rol], [CN05], [Nic04]. The median structure extends nicely to the Roller compactification.

We define the interval:

I(v,w):={m∈¯¯¯¯¯X:Uv∩Uw⊂Um}.

In the special case that v,w∈X this is the collection of vertices that are crossed by an edge geodesic connecting v and w.

Then, the fact that ¯¯¯¯¯X is a median space2
is captured by the following: for every u,v,w∈¯¯¯¯¯X there is a unique m∈¯¯¯¯¯X such that

{m}=I(u,v)∩I(v,w)∩I(w,u).

This unique point is called the median of u, v,
and w and will sometimes be denoted by m(u,v,w). In terms of half-spaces, we have:

Um=(Uu∩Uv)∪(Uv∩Uw)∪(Uw∩Uu),

which is captured by this beautiful Venn diagram:

Uv

Uu

Uw

.

While general CAT(0) cube complexes can be quite wild,3 the structure of intervals is tamable by the following (see
[BCG+09, Theorem 1.16]):

Theorem 2.7.

[BCG+09, Theorem 1.16] Let v,w∈¯¯¯¯¯X.
Then the vertex interval I(v,w) isometrically embeds into ¯¯¯¯ZD
(with the standard cubulation) where D is the dimension of X.

The proof of this employs Dilworth’s Theorem which states that a partially ordered set has finite width D if and only if it can be partitioned into D-chains. Here the partially ordered set is Uw∖Uv. Set inclusion yields the partial order and an antichain corresponds to a set of pairwise transverse half-spaces. By reversing the chains of half-spaces in Uw∖Uv in a consistent way, we may find other pairs x,y∈¯¯¯¯¯X such that I(x,y)=I(v,w). This yields the following:

Corollary 2.8.

If X has dimension D, then for any interval I⊂¯¯¯¯¯X, there are at most 2D elements on which I is an interval.

2.5. Projections and Lifting Decompositions

It is straightforward, thanks to Theorem 2.3 that if H′⊂H is an involution invariant subset, then there is a natural quotient map X(H)→X(H′). Furthermore, if H′ is Γ-invariant for some acting group Γ then the quotient is Γ-equivariant as well. One can ask to what extent this can be reversed. Namely, when is it possible to find an embedding X(H′)↪X(H)? And if H′ is assumed to be Γ-invariant, can the embedding be made to be Γ-equivariant?

Definition 2.9.

Given a subset H′⊂H(X), a lifting decomposition
is a choice of a consistent subset s⊂H(X) such that

H(X)=H′⊔(s⊔s∗).

We note that a necessary condition for the existence of a lifting decomposition is that H′ be involution invariant and that it be convex, i.e. if h,k∈H′, and h⊂ℓ⊂k then ℓ∈H′.

Given a consistent set s⊂H(X), one can associate a set of walls (viewed as an involution invariant set of half-spaces) Hs:=H(X)∖(s⊔s∗) so that s is a lifting decomposition of Hs, though there could of course be others.

The terminology is justified by:

Proposition 2.10.

Suppose that H′⊂H(X). If there exists s a lifting decomposition for H′ then there is an isometric embedding ¯¯¯¯¯X(H′)↪¯¯¯¯¯X induced from the map 2H′↪2H(X) where U↦U⊔s and the image of this embedding is

∩h∈sh⊂¯¯¯¯¯X.

Conversely, if s⊂H(X) is a consistent set of half-spaces, then, setting Hs=H(X)∖(s⊔s∗) we get an isometric embedding
¯¯¯¯¯X(Hs)↪¯¯¯¯¯X obtained as above, onto

∩h∈sh⊂¯¯¯¯¯X.

If s satisfies the descending chain condition, then the image of X(Hs) is in X.

Furthermore, if the set s is Γ-invariant then, with the restricted action on the image, the above natural embeddings are Γ-equivariant.

Remark 2.11.

We note that the projection X→X(H′) obtained by forgetting the half-spaces H∖H′ is onto. This means that if there is a lifting decomposition ¯¯¯¯¯X(H′)↪¯¯¯¯¯X then the relationship between two half-spaces (i.e. facing, transverse, etc.) is equivalent if one considers them as half-spaces in X or in X(H′).

Let us interpret the significance of Proposition 2.10 in the context of the collection of the involution-invariant set of half-spaces H(v,w):=Uv△Uw, for v,w∈¯¯¯¯¯X. These are the half-spaces separating v and w. Then, the collection of half-spaces H(v,w)+:=Uv∩Uw, i.e. those that contain both v and w, is a consistent set of half-spaces and it is straightforward to verify that H(v,w)+ is a lifting decomposition for H(v,w), yielding an isometric embedding of the CAT(0) cube complex associated to H(v,w) onto I(v,w).

There are three notions that together form a powerful framework within which to study CAT(0) cube complexes. The first is the classical notion of a nonelementary action. Caprace and Sageev showed that this allows one to study the essential core of a CAT(0) cube complex [CS11], which is the second notion. Finally, Behrstock and Charney introduced the notion of strong separation which allows for the local detection of irreducibility [BC12], which was shown by Caprace and Sageev to be available in the nonelementary setting [CS11].

3.1. Nonelementary Actions

As a CAT(0) space, a CAT(0) cube complex has a visual boundary
∂∢X which is obtained by considering equivalence classes of geodesic rays, where two rays are equivalent if they are at bounded distance
from each other. The topology on ∂∢X is the cone topology (which coincides with the topology of uniform convergence on compact subsets, when one considers geodesic rays emanating from the same base point)
[BH99]. While the visual boundary is not well behaved for non-proper spaces in general, the assumption that the space is finite dimensional is sufficient [CL10].

Definition 3.1.

An isometric action on a CAT(0) space is said to be elementary if there is a finite orbit in either the space or the visual boundary.

To exemplify the importance of this property, we have:

Theorem 3.2.

[CS11]
Suppose Γ→Aut(X) is an action on the CAT(0) cube complex X. Then either the action is elementary, or Γ contains a freely acting non-abelian free group.

3.2. Essential Actions

Caprace and Sageev [CS11] showed that for nonelementary actions, there is a nonempty “essential core” where the action is well behaved. Let us now develop the necessary terminology and recall the key facts.

Definition 3.3.

Fix a group Γ acting by automorphisms on the CAT(0) cube complex X. A half-space h∈H is called …

Γ-shallow if for some (and hence all) x∈X, the set Γ⋅x∩h is at bounded distance from h∗, otherwise, it is said to be Γ-deep.

Γ-trivial if h and h∗ are both shallow.

Γ-essential if h and h∗ are both deep.

Γ-half-essential if it is deep and h∗ is shallow.

Remark 3.4.

Observe that the collections of essential and trivial half-spaces are both closed under involution and that the collection of half-essential half-spaces is consistent. Furthermore, a half-space h∈H is Γ-essential if and only if it is Γ0-essential for any Γ0⩽Γ of finite index.

Theorem 3.5.

[CS11, Proposition 3.5]
Assume Γ→Aut(X) is a nonelementary action on the CAT(0) cube complex X then the collection of Γ-essential half-spaces is non-empty. Furthermore, if Y is the CAT(0) cube complex associated to the Γ-essential half-spaces then Y is unbounded and there is a Γ-equivariant embedding Y↪X.

The image of Y under this embedding is called the Γ-essential core. If all half-spaces are essential, then the action is said to be essential.

A simple but powerful concept introduced by Caprace and Sageev is that of flipping a half-space. A half-space h∈H is said to be Γ-flippable if there is a γ∈Γ such that h∗⊂γh.

Lemma 3.6.

[CS11, Flipping Lemma]
Assume Γ→Aut(X) is nonelementary. If h∈H is essential, then h is Γ-flippable.

Recall that a measure λ is said to be quasi-Γ-invariant whenever the following holds for every γ∈Γ, and every measurable set E: if λ(E)>0 then λ(γE)>0.

Corollary 3.7.

[CFI12]
Suppose Γ→Aut(X) is a nonelementary and essential action on the CAT(0) cube complex X. If λ is a quasi-Γ-invariant probability measure on ¯¯¯¯¯X then λ(h)>0 for every half space h∈H(X).

Proof.

Let h∈H. Then λ(h⊔h∗)=1 which means that either λ(h)>0 or λ(h∗)>0. If λ(h∗)>0 then apply the Flipping Lemma 3.6 and deduce that there is a γ∈Γ such that h∗⊊γh and hence λ(γh)⩾λ(h∗)>0. But of course, λ is Γ-quasi-invariant so λ(h)>0.
∎

Another very important operation on half-spaces developed by Caprace and Sageev is the notion of double skewering:

Lemma 3.8.

[CS11, Double Skewering]
Suppose Γ→Aut(X) is a nonelementary action on the CAT(0) cube complex X. If h⊊k are two essential half spaces, then there exists a γ∈Γ such that

γk⊊h⊊k.

The following is almost a direct consequence of the definitions. The reader will find a more in depth formulation in [CS11, Proposition 3.2]

Lemma 3.9.

If Γ acts on the CAT(0) cube complex X and preserves a finite collection of half-spaces then the Γ-action is either elementary or not essential.

An action of Γ on X is said to be Roller nonelementary if there is no finite orbit in the Roller compactification. Of course, having a finite orbit in X is equivalent to having a fixed point, and so, what distinguishes the Roller nonelementary from the visual nonelementary actions is the existence of finite orbits in the corresponding boundaries. Furthermore, (visual) nonelementary actions are necessarily Roller nonelementary, though the converse is false in general. One can take for example the standard action of Γ=Z×F2 on Z×T, where T is the standard Cayley tree. It is straightforward to see that this example is essential and elementary but not Roller elementary. On the other hand, if we set ∂Z={−∞,∞} then both {∞}×T and {−∞}×T are Γ-invariant and nonelementary. This phenomenon is captured in the following:

Proposition 3.10.

([CFI12, Proposition 2.26], [CS11]) Let X be a finite dimensional CAT(0) cube complex
and let Γ→Aut(X) be an action on X. One of the following hold:

The Γ-action is Roller-elementary.

There is a finite index subgroup Γ′<Γ and a Γ′-invariant
subcomplex ¯¯¯¯¯X′↪¯¯¯¯¯X associated to a Γ′-invariant H′⊂H(X) on which the Γ′-action is nonelementary and essential.

Moreover, if the action of Γ is nonelementary on X then X′↪X and X′ is the Γ-essential core.

3.3. Product Structures

A CAT(0) cube complex is said to be reducible if it can be expressed as a nontrivial product. Otherwise, it is said to be irreducible. A CAT(0) cube complex X with half-spaces H, admits a product decomposition X=X1×⋯×Xn if and only if there is a decomposition

H=H1⊔⋯⊔Hn

such that if i≠j then hi⋔hj for every (hi,hj)∈Hi×Hj and Xi is the CAT(0) cube complex on half-spaces Hi.

Remark 3.11.

This means that an interval in the product is the product of the intervals. Namely if (x1,…,xn),(y1,…,yn)∈¯¯¯¯¯X1×…¯Xn then

I((x1,…,xn),(y1,…,yn))=I(x1,y1)×⋯×I(xn,yn).

The irreducible decomposition is unique (up to permutation of the factors) and Aut(X) contains Aut(X1)×⋯×Aut(Xn) as a finite index subgroup. Therefore, if Γ acts on X by automorphisms, then there is a subgroup of finite index which preserves the product decomposition
[CS11, Proposition 2.6].

We take the opportunity to record here that the Roller boundary is incredibly well behaved when it comes to products:

∂X=n⋃i=1¯¯¯¯¯¯¯X1×…¯¯¯¯¯¯¯¯¯¯¯Xj−1×∂Xj×¯¯¯¯¯¯¯¯¯¯¯Xj+1×⋯×¯¯¯¯¯¯¯Xn.

While the definition of (ir)reducibility for a CAT(0) cube complex in terms of its half-space structure is already quite useful, its global character makes it at times difficult to implement. Behrstock and Charney developed an incredibly useful notion for the Salvetti complexes associated to Right Angled Artin Groups, which was then extended by Caprace and Sageev.

Two half-spaces h,k∈H are said to be strongly separated if there is no half-space which is simultaneously transverse to both h and k. For a subset H′⊂H we will say that h,k∈H′ are strongly separated in H′ if there is no half-space in H′ which is simultaneously transverse to both h and k.

The following is proved in [BC12] for (the universal cover of) the Salvetti complex of non-abelian RAAGs:

Theorem 3.13.

[CS11] Let X be a finite dimensional irreducible CAT(0) cube complex such that the action of Aut(X) is essential and non-elementary. Then X is irreducible if and only if there exists a pair of strongly separated half-spaces.

3.4. Euclidean Complexes

Definition 3.14.

Let X be a CAT(0) cube complex.
We say that X is Euclidean if the vertex set with the combinatorial metric
embeds isometrically in ZD with the ℓ1-metric, for some D<∞.

As our prime example of a Euclidean CAT(0) cube complex is an interval, which is the content of Theorem 2.7.

Definition 3.15.

An n-tuple of half-spaces h1,…,hn∈H is said to be facing if for each i≠j

h∗i∩h∗j=∅.

As an obstruction to when a CAT(0) cube complex is Euclidean, there is the following:

Lemma 3.16.

[CFI12, Lemma 2.33] If X is a Euclidean CAT(0) cube complex
that isometrically embeds into ZD, then any set of pairwise facing halfspaces
has cardinality at most 2D.

The following is an important characterization of when a complex is Euclidean.

Corollary 3.17.

[CFI12, Corollary 2.33], [CS11] Let X be an irreducible finite dimensional CAT(0)
cube complex so that the action of Aut(X) is essential and non-elementary. The following are equivalent:

X is an interval;

X is Euclidean;

H(X) does not contain a facing triple of half-spaces.

Remark 3.18.

The statement of [CFI12, Corollary 2.33] states that X is Euclidean if and only if H(X) does not contain a facing triple. However, the proof actually shows that (2) implies (3) and (3) implies (1). The missing (1) implies (2) is of course provided by Theorem 2.7[BCG+09].

Lemma 3.19.

Let X be an interval on v,w∈¯¯¯¯¯X. Then Aut(X) is elementary.

Proof.

If X is an interval then the collection of points on which it is an interval is finite and bounded above by 2D by Corollary 2.8. Let Γ0 be the finite index subgroup which fixes this set pointwise and let v belong to this set. Then, for every finite collection h1,…,hn∈Uv the intersection n∩i=1hi is not empty. Hence, the intersection of the visual boundaries corresponding to the hi must be non-empty and its unique circumcenter is fixed for the Γ0-action by [CS11, Proposition 3.6].
∎

Lemma 3.20.

[CFI12, Lemma 2.28]. Let Γ→Aut(X) be a non-elementary action.
Then the Γ0-action on the irreducible factors of the essential core is
also non-elementary and essential, where Γ0 is the finite index subgroup preserving this decomposition.

Corollary 3.21.

If Γ→Aut(X) is nonelementary then any irreducible factor in the essential core of X is not Euclidean and hence not an interval.

3.5. The Combinatorial Bridge

Behrstock and Charney showed that the CAT(0) bridge connecting two strongly separated walls is a finite geodesic segment [BC12]. In [CFI12] this idea is translated to the “combinatorial”, i.e. median setting. Most of what follows is from or adapted from [CFI12], though the notation differs slightly. Recall our convention that, {k,k∗} is denoted by ^k, for a half-space k, and that given another half-space h we will say that ^k⊂h if either k or k∗ is a proper subset of h.

Remark 3.22.

Observe that for two half-spaces h,k we have that h∩k and h∗∩k are both nonempty if and only if h⋔k or ^h⊂k.

Definition 3.23.

Let h1⊊h∗2 be a nested pair of halfspaces and β(h1,h2) denote the collection of half-spaces h∈H such that one of the following conditions hold:

^h1⊂h and h2⋔h;

^h2⊂h and h1⋔h;

^h1,^h2⊂h.

Furthermore, a half-space h will be said to be of type (1), (2), or (3) if it satisfies the corresponding property.

We note that both h1 and h2 are not of types (1)–(3). Furthermore, since h1 and h2 are disjoint, condition (1) actually means that h1⊂h and h2⋔h (and analogously for condition (2)).

Lemma 3.24.

Proof.

We begin by observing that if h∈β(h1,h2) then we necessarily have that h∗∉β(h1,h2).

Now suppose that h∈β(h1,h2) is of type (3). If h⊂k then clearly k is also of type (3) and hence k∈β(h1,h2).

Next suppose that h is of type (1) and h⊂k. Then, ^h1⊂k. Since h2⋔h and h⊂k we have that h2∩k and h∗2∩k are both nonempty. By Remark 3.22, either k⋔h2 or ^h2⊂k, and so k∈β(h1,h2).

Of course, a symmetric argument shows that if h of type (2) and h⊂k then k∈β(h1,h2).

Next we turn to the question of the descending chain condition. Since there are finitely many half-spaces in between any two, an infinite descending chain will eventually fail to satisfy all three conditions (1) through (3).
∎

Definition 3.25.

The (combinatorial) bridge between h1⊊h∗2 is denoted by B(h1,h2) and corresponds to ∩h∈β(h1,h2)h⊂X.

Lemma 3.26.

Assume that h1⊊h∗2 and set β=β(h1,h2). The collection H′=H∖(β⊔β∗) consists of half-spaces h such that one of the following hold:

h⋔h1 and h⋔h2;

up to replacing h by h∗ we have that

h1⊆h⊆h∗2.

Proof.

It is clear that if h is a half-space that is transverse to both h1 and h2 then h∉β⊔β∗.
It is also clear that if ^h=^h1 or ^h=^h2 then h∉β⊔β∗.

Now assume that h1⊊h⊊h∗2. Then h⊃^h1 and h does not contain nor is it transverse to ^h2 and hence h∉β⊔β∗.

Conversely, suppose that h∉β⊔β∗. If h⋔h1 then since h and h∗ are not type (2) we must have that h⋔h2. Assume then that h is not transverse to both h1 and h2.

If either ^h=^h1 or ^h=^h2 then up to replacing h by h∗ we have that h1⊆h⊆h∗2. Therefore, suppose that ^h≠^h1 and ^h≠^h2. Then, each of ^h1 and ^h2 is contained in h or h∗. Since both h and h∗ are not of type (3), we must have that, up to replacing h by h∗, ^h1⊂h and ^h2⊂h∗. Now, of course, since h1∩h2=∅, we conclude that h1⊂h and h2⊂h∗, i.e.

h1⊂h⊂h∗2.

∎

Corollary 3.27.

Assume h1⊂h∗2 are strongly separated. With the notation as in Lemma 3.26, β gives a lifting decomposition of H′. Furthermore, there exists a unique xi∈hi such that

B(h1,h2)=I(x1,x2).

Proof.

The fact that β is a lifting decomposition for

H′={h∈H:h1⊆h⊆h∗2 or h1⊆h∗⊆h∗2}

follows from Lemmas 3.24 and 3.26. In particular, H′ is precisely the set of half-spaces which separate points in B(h1,h2).

Let us show that B(h1,h2) is an interval. To this end, let Si=hi∩B(h1,h2). Since hi∈H′ it follows that Si≠∅.

Fixing i, suppose that x,y∈Si. Then, any wall separating them must belong to H′. By Lemma 3.26 and the assumption that h1 and h2 are strongly separated, (again replacing h by h∗ if necessary) we see that h1⊂h⊂h∗2. This means of course that h1∩h∗=∅ and hence x=y, i.e. Si is a singleton, for both i=1,2.

Set Si={xi}. Once more, since h1 and h2 are strongly separated, the collection H′ corresponds to half-spaces nested in between h1 and h∗2 and hence H′⊂H(x1,x2). Conversely, if h∈H(x1,x2) then h separates the two points x1,x2∈B(h1,h2) and hence h∈H′.
∎

Lemma 3.28.

Assume that h1⊂h∗2 are strongly separated. If ξi∈hi⊂¯¯¯¯¯X, and p∈B(h1,h2), then

m(ξ1,p,ξ2)=p.

Proof.

Let m=m(ξ1,p,ξ2). Recall that m is uniquely determined by

Um=(Uξ1∩Up)∪(Up∩Uξ2)∪(Uξ2∩Uξ1),

and so we must show that if h∈Uξ2∩Uξ1 then h∈Up. In fact, we will show that if ξ1,ξ2∈h then h∈β(h1,h2)⊂Up.

By assumption ξi∈h∩hi≠∅. Furthermore, since h1 and h2 are strongly separated, h can not be transverse to both h1 and h2. Suppose that h is parallel to h2. Since, ξ2∈h2∩h and ξ1∈h∗2∩h by Remark 3.22 we have that h⊃^h2. The same argument shows that either h is transverse to h1 or contains ^h1 and therefore h∈β(h1,h2).
∎

3.6. More Consequences

Lemma 3.29.

[CFI12, Lemma 2.28] Suppose that Γ→Aut(X) is a nonelementary and essential action, with X irreducible.

If h∈H then there exists γ,γ′∈Γ such that the following are pairwise strongly separated

γh⊂h⊂γ′h.

In each orbit, there are n-tuples of facing and pairwise strongly separated half-spaces.

Lemma 3.30.

Let X be an irreducible CAT(0) cube complex with a nonelementary and essential Γ-action. Let h∈H and n∈Z with n⩾2. Then, there exists an n-tuple {k1,…,kn} contained in a single Γ-orbit that are facing and pairwise strongly separated such that

^h∈n∩i=1ki.

Proof.

Now, assume n>2. Let {b1,…,bn+1} be the collection of facing and pairwise strongly separated half-spaces guaranteed by Item (2) of Lemma 3.29. For each i=1,…,n+1 exactly one of the following possibilities hold:

^h=^bi;

h⋔bi;

^h⊂b∗i;

^h⊂bi.

Furthermore, since the collection is strongly separated and facing, there is at most one i, assume it is i=n+1, for which the mutually exclusive items (a) through (c) can occur. Therefore, we have that

^h⊂n∩i=1bi.

Finally, if the constructed set does not belong to the same orbit, one may skewer and flip to assure that they do belong to the same orbit yielding the desired collection.
∎

Lemma 3.31.

Suppose that X is an irreducible CAT(0) cube complex with a nonelementary and essential action of the group Γ. Any nonempty subset H′⊂H verifying the following properties must be equal to H:

(Symmetric):(H′)∗=H′;

(Γ-invariant):Γ⋅H′=H′;

(Convex): If h,h′∈H′ with h⊂h′ and k∈H such that h⊂k⊂h′ then k∈H′.

Proof.

Since X is irreducible, and H′ is nonempty and Γ-invariant, we can apply Lemmas 3.29 and 3.8 to obtain a bi-infinite sequence of pairwise strongly separated half-spaces {hn:n∈Z}⊂H′ with hn+1⊂hn.

Let k∈H. Then, there is at most one element of {hn:n∈Z} which is transverse to k. This means that, there is an N∈Z for which ^k⊂h∗N+2∩hN. Since H′ is symmetric and convex, we conclude that k∈H′.
∎

Corollary 3.32.

Assume we have an essential and nonelementary action of Γ on X, and Γ0⩽Γ of finite index. If H′⊂H is a non-empty symmetric convex Γ0-invariant collection of half-spaces. Then either H′=H or X≅X′×X′′ and H′ is the half-space structure for X′.

The interested reader should consult the following references for further details [Fur02], [Kai03], [BS06], [CFI12], [BF14]. This exposition follows closely these sources, as well as a nice series of lectures by Uri Bader at CIRM in winter 2014.

Definition 4.1.

Consider a measurable action α:Γ×M→M of the group Γ on the measure space (M,m) and μ a measure on Γ. The convolution as a measure on M is the push forward under the action map of the product measure from Γ×M:

μ∗m=α∗(μ⊗m).

We shall make use of the following elementary fact:

Lemma 4.2.

Let Haar denote the counting measure on Γ, δe the Dirac measure at the identity e∈Γ and μ∈P(Γ) be a probability measure. Then Haar∗μ=Haar, and δe∗μ=μ=μ∗δe.

The proof of this is straightforward but we record it to exemplify the usefulness of thinking of convolution of measures in the context of pushforwards as above:

Proof.

Let us show that Haar∗μ(γ)=1 for every γ∈Γ. Indeed,

Haar∗μ(γ)

=

Haar⊗μ{(γ0,γ1):γ0γ1=γ}

=

∑γ1∈ΓHaar(γγ−11)μ(γ1)

=

∑γ1∈Γμ(γ1)=1

A similar calculation shows that δe∗μ=μ=μ∗δe.
∎

Definition 4.3.

A probability measure μ∈P(Γ) is said to be generating if for every γ∈Γ there are hi∈supp(μ) such that γ=h1⋯hn, i.e. the support of μ generates Γ as a semigroup.

Given a generating measure μ, we will associate two spaces to the μ-random walk. The space of increments and the path space. As sets, these two spaces will be the same, but the measures on them will be different.

Let ΓN={¯¯¯ω=(ωn)n⩾1:ωn∈Γ}. The measure μ on Γ naturally induces a measure μN on ΓN which assigns measure μ(g1)⋯μ(gn) to the cylinder set:

Extra open brace or missing close brace

Let Ω:=Γ×ΓN={(ω0,ω1,…):ωn∈Γ}.
Given another measure θ on Γ, which is not assumed to be a probability measure, we can consider the associated measure θ⊗μN on Ω. This is the space of increments, where we see the first factor as where to start the random walk (with distribution θ). We will consider the action of Γ on Ω which is transitive on the first factor and trivial on the rest.

Next let Ω′=ΓN. We will consider the diagonal action of Γ on Ω′.
Observe that there is a natural map W:Ω→Ω′, (ω0,ω1,ω2,…)↦¯¯¯ω′ where the n-th component of the image is given by

ω′n=ω0ω1ω2⋯ωn−1.

With the actions of Γ defined above on Ω and Ω′ we note that W is Γ-equivariant.
We think of the image of this map as the space of sample paths.
Consider the time shift map:

S:Ω→Ω;(ω0,ω1,ω2…)↦(ω0ω1,ω2,ω3,…)

which is just a composition of the standard action map Γ×Γ→Γ given by (ω0,ω1)↦ω0ω1 with the time shift map: